# Development and Application of London Dispersion Corrections for Electronic Structure Methods

## Authors

- Eike Caldeweyher

## Abstract

Mean-field electronic structure methods like Hartree–Fock (HF) or Kohn–Sham (KS) Density Functional Theory (DFT) do not account for long-range correlation effects and consequently not for London Dispersion (LD). These LD forces contribute significantly to the interaction between molecular aggregates and are thus mandatory for a quantitative comparison of in silico predictions with experimental observations. Casimir and Polder formulated long-range correlation in terms of dynamic polarizabilities and established the foundation of all LD corrections within a DFT framework. This dissertation develops an efficient correction scheme, termed DFT-D4, for the accurate calculation of LD correlation effects in combination with Density Functional Approximations (DFAs) or other mean-field electronic structure methods. The presented scheme is an extension of the well-established DFT-D3 LD correction. In DFT-D3, the Coulomb operator is expanded into multipoles where a coarse-grain partitioning to atomic polarizabilities enables the calculation of interatomic dipole-dipole dispersion coefficients. Fractional Coordination Numbers (CNs) are used as a measure of the hybridization to efficiently calculate hybridization dependent dispersion coefficients from dynamic Atom-in-Molecule (AIM) polarizabilities. In order to better account for non-additive AIM-polarizability effects, DFT-D4 additionally uses atomic oxidation state information. Those oxidation state information are of particular importance in systems like organometallic or charged complexes and improve their description of noncovalent interactions substantially. The oxidation-state dependency is implemented by an empirical function which uses atomic charge information for the scaling of reference polarizabilities. This scaling procedure is shown to be well suited to incorporate the most significant electronic effects into the reference values. The DFT-D4 default method uses a classical charge model, however, other charge schemes are implemented as well. A D3-like interpolation scheme incorporates those scaled references and generates hybridization and oxidation state-dependent dynamic polarizabilities. DFT-D4 is shown to yield excellent molecular polarizabilities and dispersion coefficients. On a database of 1225 intermolecular dispersion coefficients, the Mean Absolute Deviation (MAD) from experimental references is 3.8%. When combined with appropriate DFAs, noncovalent interaction energies for large complexes have MADs well below 10% and rotational constants (measure for the molecular size) have the accuracy of high-level correlated methodologies. For certain metal-ions in highly polar and periodic environments, DFT-D methods obtain too large atomic polarizabilities compared to Time-Dependent Density Functional Theory (TD-DFT) values (e. g., Na cation in crystalline NaCl). A more in-depth analysis shows that the absence of suitable reference systems causes the identified problem. In periodic environments, the CNs quickly approach values far beyond those for which reference polarizabilities are available. This absence leads to CNs extrapolations of polarizabilities, which are not reliable. The periodic DFT-D4 removes this drawback by extending the scope of reference polarizabilities for highly coordinated systems. For this purpose, dynamic polarizabilities for pseudo-periodic references with high CNs are calculated. The addition of such references to the D4-reference pool enables the physically reasonable calculation of atomic polarizabilities in dense solids. Such improved polarizabilities are shown to be beneficial for obtaining high-quality adsorption energies. Comparing the computational costs of several dispersion corrections shows large differences in terms of efficiency. While the computational costs of the DFT-D4 method are negligible with respect to the underlying DFT calculation, some correction methods become the computational bottleneck. The second part of the thesis consists of two application studies of LD corrected DFT methods. In the first case study, LD driven packing effects lead to the shortest intermolecular H· · ·H contact reported to date. The attractive interactions between tBu groups cause this unusual binding situation. A periodic dispersion corrected DFT composite scheme verifies the experimental finding and an energy decomposition analysis quantifies the importance of LD interactions. The DFT structure determined by this composite scheme agrees very well with the structure determined from a low-temperature neutron diffraction experiment (intermolecular hydrogen-hydrogen bond length deviate by ≈1 pm from the experimental one). The second case study investigates LD interactions in organometallic complexes. Properly accounting for LD is shown to be important for predicting reaction paths, e. g, for the design of dispersion-controlled reaction sequences in homogeneous catalysis. In other systems like in the [[Rh(CNPh)4]2]2+ dication, LD contributions are able to compensate substantial repulsive electrostatic interactions. Additionally, LD interaction energies are compared to values obtained by a Local Energy Decomposition (LED) employing a local coupled-cluster theory. In combination with certain DFAs, the LD energies are in good agreement with the values of the LED partitioning. In summary, the DFT-D4 LD correction is recommended as a standard tool in computer-assisted chemistry of molecular and periodic systems due to its high accuracy and computational efficiency.